微积分练习题和答案1

精编WORD文档下载可编缉打印下载文档，远离加班熬夜微积分练习题和答案1篇一：微积分1期末模拟考试答案(1)微积分1期末模拟考试一、填空题（每题3分，共18分）ex?11．li?；x?0?x2.?()dx?xsin?x；c3.设函数f(x)?(x?1)(x?2)(x?3)，则f(x)有_______个极值点;4.设f(x)的一个原函数为e，则?f(x)dx?，?f?(x)dx?。?x5.函数f(x)?arctanx在?0,1?上满足拉格朗日中值定理的点??。6.设f(x)=sinx,则?f??2x?dx??11?7.??2?x?a(x?a)??dx＝.(附加题，可不做)?精编WORD文档下载可编缉打印下载文档，远离加班熬夜8.99(2x?3)dx?.(附加题，可不做)?二、选择题（每题3分，共30分）1.设?f(x)dx?A、1?x?C，则f(x)?（B）1?x?22?2x2xB、C、Ｄ、(x?1)2(x?1)2(x?1)2(x?1)22.设f?(x)连续，下列等式错误的是（D）A、C、???f(x)dx?f(x)B、??f?(x)dx?f(x)?C??f(2x)dx???f(2x)D、?f?(2x)dx?f(2x)?C3.设?f(x)dx?C，则?xf(x2)dx?（A）A、1sinx?CB、1C22C、1sin2CD、1sin2x?C2x精编WORD文档下载可编缉打印下载文档，远离加班熬夜4.设?xf(x)dx?sinex?C，则?f(lnx)?（C）A、sine?CB、sin(lnx)?C5.设函数f(x)在x?x0处取得极大值，则必有（D）。A.f?(x0)=0；C.f?(x0)?0；B.f?(x0)?0；D.f?(x0)=0或者f?(x0)不存在.6.函数f(x)?lnx及其图形在区间(1,??)上(B)。A.单调减少上凹;B.单调增加上凸;C.单调减少上凸;D.单调增加上凹.7.如果f(x)?(A),那么f?(x)?0。A.arcsinx?arccosx；C.sinx?cosx；B.sec2x?tanx；D.lnx?arccosx.8.函数y?x?arctanx在(??,??)内（A）A.单调递增B.单调递减C.不单调D.不连续9.设f(x)?x，则x?0是f(x)的（D）精编WORD文档下载可编缉打印下载文档，远离加班熬夜23A.间断点B.可导点C.驻点D.极值点10.（C）A.asintB.atantC.asectD.acost三、解下列各题（每小题4分，共8分）ex?e?x?21.求lim；x?01?cosx?1ln(1?x)??2.求lim；x?0?xx2?1x3.求lim(cosx)x?02.(附加题，可不做)四、求不定积分（每小题5分，共30分）;1.?1?x2.;精编WORD文档下载可编缉打印下载文档，远离加班熬夜3.34x(2x?e?cos5x)dx;?4.2x?lnxdx;5.?;6.dx?x(1?x2);7.;(附加题，可不做)8.;(附加题，可不做)1?x?x2;(附加题，可不做)9.?2(1?x)210.2sin?精编WORD文档下载可编缉打印下载文档，远离加班熬夜(附加题，可不做)11.设函数f(x)的一个原函数为lnx，求不定积分xf?(x)dx。(附加题，可不做)?篇二：微积分课后习题答案习题1—1解答1．设f(x,y)?xy?x11x1，求f(?x,?y),f(,),f(xy,),yxyyf(x,y)11xy1xyyxxy22解f(?x,?y)?xy?xy；f(,)??;f(xy,)?x?y;1f(x,y)?yxy2精编WORD文档下载可编缉打印下载文档，远离加班熬夜?x2．设f(x,y)?lnxlny，证明：f(xy,uv)?f(x,u)?f(x,v)?f(y,u)?f(y,v)f(xy,uv)?ln(xy)?ln(uv)?(lnx?lny)(lnu?lnv)?lnx?lnu?lnx?lnv?lny?lnu?lny?lnv?f(x,u)?f(x,v)?f(y,u)?f(y,v)3．求下列函数的定义域，并画出定义域的图形：（1）f(x,y)??x2?2y?1;2（2）f(x,y)?4x?y2ln(1?x?y)xa222;（3）f(x,y)?1??精编WORD文档下载可编缉打印下载文档，远离加班熬夜yb22?zc22;（4）f(x,y,z)?x?2y?2z2.?x?y?z解（1）D?{(x,y)x?1,y?1?（2）D?(x,y)0?x2?y2?1,y?222??xy（3）精编WORD文档下载可编缉打印下载文档，远离加班熬夜D??(x,y)2?2?ab?（4）D?(x,y,z)x?0,y?0,z?0,x2?y2?z2?14．求下列各极限：（1）lim1?xyx?y22??1?00?1?1x?0y?1=（2）limln(x?ex?y2?2y)2x?1y?0?ln(1?e)?0(2??ln2（3）limxy?4xy精编WORD文档下载可编缉打印下载文档，远离加班熬夜x?0y?0?limxy?4)(2?xy(2?xy?4)x?0y?0xy?4)??14（4）limsin(xy)yx?2y?0?limsin(xy)xyx?2y?0?x?25．证明下列极限不存在：（1）limx?0y?0x?yx?y;（2）limxy22精编WORD文档下载可编缉打印下载文档，远离加班熬夜222x?0y?0xy?(x?y)（1）证明如果动点P(x,y)沿y?2x趋向(0,0)则limx?yx?yx?2xx?2x??3；x?0y?2x?0?limx?0如果动点P(x,y)沿x?2y趋向(0,0)，则limx?yx?yy?0x?2y?0?lim3yyy?0?3所以极限不存在。（2）证明如果动点P(x,y)沿y?x趋向(0,0)精编WORD文档下载可编缉打印下载文档，远离加班熬夜xy22222则limx?0y?x?0xy?(x?y)?limxx44x?0?1；如果动点P(x,y)沿y?2x趋向(0,0)，则lim所以极限不存在。6．指出下列函数的间断点：（1）f(x,y)?y?2xy?2x2xy2222精编WORD文档下载可编缉打印下载文档，远离加班熬夜2x?0y?2x?0xy?(x?y)?lim4x442x?04x?x?0；（2）z?lnx?y。解（1）为使函数表达式有意义，需y2?2x?0，所以在y2?2x?0处，函数间断。（2）为使函数表达式有意义，需x?y，所以在x?y处，函数间断。习题1—21．（1）z??z?xxy?yx,?z?x精编WORD文档下载可编缉打印下载文档，远离加班熬夜?1y?yx2，?z?y?1x?xy2.(2)?ycos(xy)?2ycos(xy)sin(xy)?y[cos(xy)?sin(2xy)]?z?y?xcos(xy)?2xcos(xy)sin(xy)?x[cos(xy)?sin(2xy)](3)?z?x?y(1?xy)y?1y?y(1?xy)2y?1精编WORD文档下载可编缉打印下载文档，远离加班熬夜,1?zz?yx1?xy,lnz=yln，两边同时对y求偏导得?ln(1?xy)?y?z?y?z[ln1(?xy)?xy1?xy]?(1?xy)[ln1(?xy)?yxy1?xy];(4)?z?x1??x?2yxxy31y2?x?2yx(x?y)精编WORD文档下载可编缉打印下载文档，远离加班熬夜33?z,?y?xx?2yx2?;3x?y1?u(5)?x?yzxz?1,?u?y精编WORD文档下载可编缉打印下载文档，远离加班熬夜?1zyxzlnx,?u?z??yz2yxzlnx;(6)?u?x?z(x?y)z?12z1?(x?y),精编WORD文档下载可编缉打印下载文档，远离加班熬夜?u?y??z(x?y)z?12z1?(x?y),?u?z?(x?y)ln(x?y)1?(x?y)2zz;2.(1)zx?y,zy?x,zxx?0,zxy?1,zyy?;(2)zx?asin2(ax?by),zy?bsin2(ax?by),zxx?2acos2(ax?by),zxy?2abcos2(ax?by),zyy?2bcos2(ax?by).223fx?y?2xz,fy?2xy?z,fz?2yz?x,fxx?2z,fxz?2x,fyz?2z,精编WORD文档下载可编缉打印下载文档，远离加班熬夜fxx(0,0,1)?2,fxz(1,0,2)?2,fyz(0,?1,0)?0.2224zx??2sin2(x?t2),zt?sin2(x?t2yt2),zxt?2cos2(x?t2)?0.t2),ztt??cos2(x?t2)2ztt?zxt??2cos2(x?)?2cos2(x?5.(1)zx??12yx2y精编WORD文档下载可编缉打印下载文档，远离加班熬夜e,zy?x1xex,dz??yx2yedx?x1xyexdy;(2)z?ln(x2?y),zx?2xx?y22,zy?yx?y2精编WORD文档下载可编缉打印下载文档，远离加班熬夜2,dz?xx?y22dx?yx?y22dy;(3)zx2yx???2,zy?2y2x?y1?()1?x?y1?ydx?xdyxx?2dz?,;222y2x?yx?y()xyz精编WORD文档下载可编缉打印下载文档，远离加班熬夜(4)ux?yzxyz?1,uy?zxyz?1yzlnx,uz?yxyzlnx,lnxdz.du?yzxdx?zxlnxdy?yxyz6.设对角线为z,则z?22x?y,zx?xx?y22,zy?yx?y22,dz?xdx?ydyx?y精编WORD文档下载可编缉打印下载文档，远离加班熬夜22当x?6,y?8,?x?0.05,?y??0.1时,?z?dz?6?0.05?8?(?0.1)6?822=-0.05(m).7.设两腰分别为x、y,斜边为z,则z?zx?xx?y22x?y,22，zy?yx?y22,dz?xdx?ydyx?y22精编WORD文档下载可编缉打印下载文档，远离加班熬夜,设x、y、z的绝对误差分别为?x、?y、?z,当x?7,y?24,?x??x?0.1,?y???z?dz?7?0.1?24?0.17?2422y?0.1时,z?7?2422?25=0.124,z的绝对误差?z?0.124z的相对误差?zz?0.12425?0.496%.8.设内半径为r，内高为h，容积为V，则V??rh,Vr?2?rh,Vh??r,dV?2?rhdr??rdh,222精编WORD文档下载可编缉打印下载文档，远离加班熬夜当r?4,h?20,?r?0.1,?h?0.1时,?V?dV?2?3.14?4?20?0.1?3.14?4?0.1?55.264(cm).23习题1—3yx?)21.dudx??fdx?xdx??fdy?ydx??fdz?zdx?1?(zxyzax41?(精编WORD文档下载可编缉打印下载文档，远离加班熬夜22zxyz?xyzxyz2?ae)2ax?1?(?2a(ax?1))2=y[z?axz?2axy(ax?1)]z?xy??f?????x3222=精编WORD文档下载可编缉打印下载文档，远离加班熬夜(ax?1)e(1?ax)22ax(ax?1)?xe.342.?z?x??f?????x=???222?x?x?y22?arcsin??4x4精编WORD文档下载可编缉打印下载文档，远离加班熬夜x?y=4xarcsin4?x?y4x?y?z?y??f?????y??xln(x?y)(1?x?y)(x?y)2222444y434?f?????y精编WORD文档下载可编缉打印下载文档，远离加班熬夜2=???2?y?x?y422?arcsin??x?y=4yarcsin43?x?y42x?y?yln(x?y)(1?x?y)(x?y)22精编WORD文档下载可编缉打印下载文档，远离加班熬夜224.3.(1)?u?x?u?x?u?x=2xf1?yexyf2,?u?y=?2yf1?xe1zxyf2.(2)=1y?f1,?u?y=?xy2?